For the following exercises, determine whether the vector field is conservative and, if it is, find the potential function. 106. F(x, y) = (6x + 5y)i + (5x + 4yj
Answers
f(x, y) = 3x + 2y + C
To know if a vector field is conservative, we have to check if it is the gradient of a scalar function.
The gradient of the function f(x, y) is given by
∇f(x, y) = (∂f/∂x, ∂f/∂y)
If a vector field F(x, y) is the gradient of a scalar function f(x, y), then the following conditions must be satisfied:
F(x, y) = ∇f(x, y)
Whether a given vector field satisfies this condition can be checked by taking the partial derivative with respect to x and y.
∂F/∂x = ∂(6x + 5y)/∂x = 6
∂F/∂y = ∂(5x + 4y)/∂y = 4
The partial derivative of a given vector field is the constant, the gradient of the scalar function. So vector fields are conservative and we can find a potential function f(x, y) such that F(x, y) = ∇f(x, y).
To find the potential function, we can integrate the partial derivative of F(x, y) with respect to x and y.
f(x, y) = ∫∂F/∂xdx + ∫∂F/∂ydy
= ∫6dx + ∫4dy
= 3x + 2y + C
where C is the constant of integration. So the potential function for a given vector field is
f(x, y) = 3x + 2y + C
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Related Questions
A random sample of 100 of a certain popular car model last year found that 20 had a certain minor defect in the brakes. The car company made an adjustment in the production process to try to reduce the proportion of cars with the brake problem. A random sample of 350 of this year's model found that 50 had the minor brake defect. Describe a Type I error and a Type II error in this setting, and give a possible consequence of each.
Answers
Type I error: Reject the null hypothesis H₀, when H₀ is true.
Failure to reject the null hypothesis H₀, when H₀ is untrue is a Type II mistake.
Define hypothesisA hypothesis is a proposed explanation or prediction for a phenomenon or observed phenomenon. It is a statement that suggests an explanation for an observed fact or set of facts and provides a basis for further investigation or experimentation.
Given Information
Determine the hypothesis:
Hₐ: p₁>p₂
Type I error: Reject the null hypothesis H₀, when H₀ is true.
Although being ineffectual, the adjustment gives the impression of being effective.
As a result, more vehicles than anticipated have brake problems, which might result in additional expenses for the company.
Failure to reject the null hypothesis H₀, when H₀ is untrue is a Type II mistake.
Interpretation: The adjustment is successful even though it appears to be ineffective.
As a consequence, fewer automobiles than anticipated have brake issues, which might result in financial savings for the corporation.
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Nick's science class is observing the growth rate of a tomato plant. The tomato plant grows 3 cm every week for 12 weeks. What is the total growth of the tomato plant during these 12 weeks?
Answers
Answer:
36 cm
Step-by-step explanation:
Because the plant is growing the same amount each week you would take 12(3) and get 36 cm at the end of the 12 weeks.
What was the percent increase of China's
population from 1970 to 1995?
56.25%
45%
156.25%
9.405%
China's Population
Population
(billions)
0.7
Year
1965
1970
1975
1980
1985
1990
1995
2000
0.8
0.92
0.98
1.08
1.14
1.25
1.26
Answers
The percentage increase China's population from 1970 to 1995 is: 56.25%.
How to Calculate Percentage Increase?Percentage increase = (Final Value - Original Value)/Original value × 100.
Given the following:
Original value = 0.8 billion
Final Value = 1.25 billion
Final Value - Original Value = 1.25 - 0.8
Final Value - Original Value = 0.45
Percentage increase = 0.45/0.8 × 100 = 56.25%.
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An equilateral triangle is folded in half.IN60° 60°14 cm-4What is x, the height of the equilateral triangle?O 14V301407307
Answers
An equilateral triangle is a triangle that has the same length on all its three sides. Therefore, we can say that:
Since the triangle is folded in half, then we can say that:
From this, we can solve "x" using the Pythagorean Theorem.
\(c^2=a^{2^{}}+b^2\)where "c" = hypotenuse and "a" and "b" can be either of the remaining sides.
\(\begin{gathered} 14^2=7^2+x^2 \\ 196=49+x^2 \\ 196-49=49+x^2-49 \\ 147=x^2 \\ \sqrt[]{147}=\sqrt[]{x^2} \\ 7\sqrt[]{3}=x \end{gathered}\)Therefore, the height of our equilateral triangle is 7√3. This is found in the third option.
Find the equation of the line with slope 7 that goes through the point
(0,8). Write your answer in slope-intercept form.
Answers
Here’s why:
The slope is 7. The point (0,8) is actually the intercept, because it gives us y when x is zero. So the intercept is 8. To write in slope intercept form is y = mx + b or y = 7x + 8.
If the required reserve ratio is 15 percent, the simple deposit multiplier is:_____
a. 1.5.
b. 15.0.
c. 6.67.
d. 3.33.
Answers
Answer:
C. 6.67
Step-by-step explanation:
Given
Reserve Ratio (RR) = 15%
Required
Determine the simple deposit multiplier
Simple deposit multiplier (M) is calculated by taking the inverse of RR.
The following formula applies:
\(M = \frac{1}{RR}\)
Substitute 15% for RR
\(M = \frac{1}{15\%}\)
Convert 15% to decimal
\(M = \frac{1}{0.15}\)
\(M=6.67\)
Hence, option C answers the question
I need help with this question
Answers
Answer:
oehfinefnefikvds
Step-by-step explanation:
okokokoko
You say goodbye to your friend at the intersection of two perpendicular roads. At time t=0 you drive off North at a (constant) speed v and your friend drives West at a (constant) speed w. You badly want to know: how fast is the distance between you and your friend increasing at time t? Enter here the derivative of the distance from your friend with respect to t: __________. Being scientifically minded you ask yourself how does the speed of separation change with time. In other words, what is the second derivative of the distance between you and your friend? __________. Suppose that after your friend takes off (at time t=0) you linger for an hour to contemplate the spot on which your friend was standing. After that hour you drive off too (to the North). How fast is the distance between you and your friend increasing at time t (greater than one hour)? __________. Again, you ask what is the second derivative of your separation: ____________.
Answers
Answer:
Let x be the distance traveled by you to the north and y be the distance traveled by your friend to the west. Let d be the distance between you and your friend at time t. Then, we have:
d^2 = x^2 + y^2
Taking the derivative of both sides with respect to time, we get:
2d (dd/dt) = 2x(dx/dt) + 2y(dy/dt)
Simplifying this expression, we get:
dd/dt = (x(dx/dt) + y(dy/dt)) / d
Since you are driving north at a constant speed v and your friend is driving west at a constant speed w, we have:
dx/dt = v and dy/dt = -w
Substituting these values and simplifying, we get:
dd/dt = (v^2 + w^2)^0.5
Therefore, the speed of separation between you and your friend at time t is the square root of the sum of the squares of your speeds.
To find the second derivative of the distance between you and your friend, we take the derivative of dd/dt with respect to time:
d^2d/dt^2 = (v dv/dt + w dw/dt) / (v^2 + w^2)^0.5
Since v and w are constant, their derivatives with respect to time are zero, so the second derivative of the distance between you and your friend is zero.
Suppose you wait for an hour (i.e., t = 1 hour) before driving off to the north. Then, at time t > 1 hour, the distance between you and your friend is given by:
d = (vt)^2 + ((t-1)w)^2)^0.5
Taking the derivative of both sides with respect to time, we get:
dd/dt = 2v^2t / (2((vt)^2 + ((t-1)w)^2)^0.5)
Simplifying this expression, we get:
dd/dt = v / ((v^2 + ((t-1)w)^2)^0.5)
Therefore, the speed of separation between you and your friend at time t (greater than one hour) is given by v divided by the square root of the sum of the squares of your speed and your friend's speed.
The second derivative of the distance between you and your friend is given by:
d^2d/dt^2 = -vw(t-1) / ((v^2 + ((t-1)w)^2)^1.5)
After working satisfactorily for 6 months, Collin will be eligible for a 7% raise. How much will Collin's
gross salary be, after her raise, for each 2-week pay period? His current gross salary is $1,083.34.
Answers
Collin's gross salary, after the 7% raise, for each 2-week pay period, will be approximately $1,159.17.
To calculate Collin's gross salary after the 7% raise for each 2-week pay period, we need to find the raise amount and add it to his current gross salary.
Collin's current gross salary is $1,083.34.
To find the raise amount, we multiply his current gross salary by 7% (or 0.07):
Raise amount = $1,083.34 \(\times\) 0.07
= $75.8338 (rounded to two decimal places)
Now, we add the raise amount to his current gross salary to find the new gross salary:
New gross salary = $1,083.34 + $75.8338 = $1,159.1738 (rounded to two decimal places)
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What is the slope of the line that passes through the points
(
−
6
,
8
)
(−6,8) and
(
−
9
,
7
)
(−9,7)? Write your answer in simplest form.
Answers
Answer:
1/3 is the slope
Answer:
1 / 3
Step-by-step explanation:
Slope = y₂ - y₁ / x₂ - x
= 7 - 8 / -9 + 6
= - 1 / - 3
= 1 / 3
A rectangular prism is filled exactly with 605 cubes. Each cube has edge length 1/5 cm.
What is the volume of the rectangular prism?
Answers
The volume of the rectangular prism is 4.84 cm cube.
How to find the volume of the rectangular prism?A rectangular prism is filled exactly with 605 cubes. Each cube has edge
length 1 / 5 cm. Therefore, the volume of the rectangular prism can be
calculated as follows:
volume of each cube = l³
volume of each cube = (1 / 5)³
volume of each cube = 1 / 125 cm³
Therefore,
volume of the rectangular prism = 605 × 1 / 125
volume of the rectangular prism = 605 / 125
volume of the rectangular prism = 4.84 cm³
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d = 9 in
Calculate the radius of the circle.
Answers
Answer:
Radius is 4.5
Step-by-step explanation:
Radius is always half the diameter.
Determine the probability of rolling a die and getting a 2
then a 5.
Answers
The probability of rolling a die and getting a 2, then a 5, is 1/36.
To determine the probability of rolling a die and getting a 2, then a 5, we need to multiply the probabilities of each event happening.
First, let's consider the probability of rolling a die and getting a 2. Since there are six equally likely outcomes when rolling a fair six-sided die (numbers 1 to 6), the probability of rolling a 2 is 1/6.
Now, let's consider the probability of rolling a die and getting a 5. Again, there are six equally likely outcomes, so the probability of rolling a 5 is also 1/6.
To find the probability of both events happening, we multiply the probabilities:
Probability of rolling a 2 and then a 5 = (1/6) * (1/6) = 1/36.
Therefore, the probability of rolling a die and getting a 2, then a 5, is 1/36.
It's important to note that each roll of the die is an independent event, meaning that the outcome of one roll does not affect the outcome of the next roll. Therefore, the probability of rolling a 2 and then a 5 remains constant at 1/36 regardless of previous rolls or the order in which they occur.
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What is the square root of -1?
Answers
Answer:
i
Step-by-step explanation:
i means imaginary because whatever you try you can't get a number which squared gets a negative number.
Triangle XYZ is rotated 90° counterclockwise about the origin to produce X'Y'Z'. What are the coordinates of X'Y'Z'?
Answers
Answer:
59
Step-by-step explanation:
6 plus 4 equals 8 plus 9
Identify the coordinates of each vertex of triangle XYZ.
Apply the 90° counterclockwise rotation rule, which states that when rotating a point 90° counterclockwise about the origin, the new coordinates of the point will be (-y, x).
Substitute the coordinates of each vertex into the rotation rule to find the coordinates of the corresponding vertex in X'Y'Z'.
For example, let's say that the coordinates of triangle XYZ are:
X = (2, 3)
Y = (-1, 4)
Z = (0, -2)
To find the coordinates of X'Y'Z', we can apply the rotation rule to each vertex as follows:
X' = (-3, 2)
Y' = (-4, -1)
Z' = (2, 0)
Therefore, the coordinates of X'Y'Z' after rotating triangle XYZ 90° counterclockwise about the origin are:
X' = (-3, 2)
Y' = (-4, -1)
Z' = (2, 0)
Note that the order of the vertices may change after the rotation, but the coordinates of each vertex will correspond to the same vertex in the original triangle.
Need help with this one step by step please? A.
500 meters
B.
360 meters
C.
63 meters
D.
1,000 meters
Answers
Answer:
B. 360 meters
Step-by-step explanation:
The triangles shown are similar, so their corresponding parts are proportional.
SetupThere are a couple of useful proportions here. We can equate the ratio of horizontal segments to the ratio of diagonal segments:
(600 m)/(250 m) = ??/(150 m)
Or, we can equate the ratios of the diagonal segment to the corresponding horizontal segment:
??/(600 m) = (150 m)/(250 m)
SolutionEither way, when we multiply the equation by the denominator of ??, we get the result ...
?? = (600)(150/250) m = 360 m
The distance from F to G is 360 meters.
choose the general algebraic representation (notation) for transforming pentagon vwxyz into pentagon qrstu.
Answers
The transformation function can be represented as vwxyz \($\rightarrow$\)qrstu = (v,w,x,y,z) \($\rightarrow$\) (q,r,s,t,u).
This algebraic representation of the transformation of pentagon vwxyz into pentagon qrstu is using the notation of a function. A function takes an input (in this case, the coordinates of pentagon vwxyz) and produces an output (in this case, the coordinates of pentagon qrstu). This transformation can be represented by the notation (v,w,x,y,z) \($\rightarrow$\) (q,r,s,t,u). This notation means that the coordinates (v,w,x,y,z) are being transformed into the coordinates (q,r,s,t,u). This notation is a concise way of expressing the transformation of pentagon vwxyz into pentagon qrstu. It is important to note that the transformation itself is not specified; the notation is simply used as a way of representing the transformation.
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A salesperson earns a commission of $252 for selling $2100 in merchandise.
Find the commission rate. Write your answer as a percentage.
Answers
Answer:
The commission rate is of 12%.
Step-by-step explanation:
Commission rate:
The commission rate is given by the commission value multiplied by 100% and divided by the value of merchandise sold.
In this question:
$252 in commission for selling $2100 in merchandise. So
252*100%/2100 = 12%
The commission rate is of 12%.
Could y’all PLEASE HELP
Answers
Answer:
should be complementary angles
4x+5(7x-3)=9(x-5)
x=???
Answers
Answer:
i think its 3 im not sure
Step-by-step explanation:
Answer:
x = -1
Step-by-step explanation:
Given equation,
→ 4x + 5(7x - 3) = 9(x - 5)
Now the value of x will be,
→ 4x + 5(7x - 3) = 9(x - 5)
→ 4x + 35x - 15 = 9x - 45
→ 39x - 15 = 9x - 45
→ 39x - 9x = -45 + 15
→ 30x = -30
→ x = -30/30
→ [ x = -1 ]
Hence, the value of x is -1.
U= {1,2,3,4,5,6,7,8,}, A={1,4,5,7}, B= {2,5,6,7,}, and C= {3,4,6,7}
Complete the following set operations:
a. AU(BUC)
b. (AN(BNC))'
c. (ANB)U(ANC)
d. (ANB')U(ANC')
Answers
Answer:
a. {1, 2, 3, 4, 5, 6, 7}
b. {1, 2, 3, 4, 5, 6, 8}
c. {4, 5, 7}
d. {1, 4, 5}
Step-by-step explanation:
The union of two sets is the list of elements in either set. The intersection of two sets is the list of elements in both sets. The complement of a set is the list of elements in the universal set that are not in the set being complemented. The complement of a set can be indicated with an apostrophe: A' is the complement of set A, for example.
a. AU(BUC)B∪C = {2, 5, 6, 7} ∪ {3, 4, 6, 7} = {2, 3, 4, 5, 6, 7}
A∪(B∪C) = {1, 4, 5, 7} ∪ {2, 3, 4, 5, 6, 7} = {1, 2, 3, 4, 5, 6, 7}
b. (AN(BNC))'B∩C = {2, 5, 6, 7} ∩ {3, 4, 6, 7} = {6, 7}
A ∩ (B∩C) = {1, 4, 5, 7} ∩ {6, 7} = {7}
(A∩(B∩C))' = {7}' = {1, 2, 3, 4, 5, 6, 8}
c. (ANB)U(ANC)A∩Β = {1, 4, 5, 7} ∩ {2, 5, 6, 7} = {5, 7}
Α∩C = {1, 4, 5, 7} ∩ {3, 4, 6, 7} = {4, 7}
(A∩B)∪(A∩C) = {5, 7} ∪ {4, 7} = {4, 5, 7}
d. (ANB')U(ANC')(A∩B')∪(A∩C') = A∩(B'∪C') = A∩(B∩C)'
(B∩C)' = {6, 7}' = {1, 2, 3, 4, 5, 8}
A∩(B∩C)' = {1, 4, 5, 7} ∩ {1, 2, 3, 4, 5, 8} = {1, 4, 5}
__
Additional comment
In part (d) we made use of De Morgan's law for sets:
B'∪C' = (B∩C)'
We also made use of the distributive property for sets:
A∩(B∪C) = (A∩B)∪(A∩C)
9(2n+1) =
can someone help me, please
Answers
Answer:
9(2n + 1) = 18n + 9Step-by-step explanation:
9(2n + 1)
use the distributive property a(b + c) = ab + ac
= (9)(2n) + (9)(1) = 18n + 9
Step-by-step explanation:
9(2n+1) =0
18n+9 = 0
18n = -9
n = -9/18
n = -1/2
Find the slope of the line that’s passes through the points (-5,18) and (1,4)
Answers
Answer:
Slope = \(\frac{7}{-3}\)
Step-by-step explanation:
(-5,18) (1,4)
\(\frac{18 - 4}{-5 - 1}\) = \(\frac{14}{-6}\) = \(\frac{7}{-3}\)
Plz help me solve this
Answers
simplify the equation, cancel equal terms, move constant to the right, calculate, divide both sides !!!!
Answer: x=-3
Step-by-step explanation:
First let's eliminate the square root
\((\sqrt{x^2-5x+2})^2=(\sqrt{x^2+17})^2\\x^2-5x+2=x^2+17\)
Now subtract \(x^2\\\) from both sides
\(x^2-5x+2-x^2=x^2+17-x^2\\-5x+2=17\)
Subtract 2 from both sides
\(-5x+2-2=17-2\\-5x=15\)
Finally divide both sides by -5
\(\frac{-5x}{-5} =\frac{15}{-5} \\x=-3\)
How do you do number 61?
Answers
The remainder of the synthetic division is 0, hence x + 2 is a factor of the polynomial
Synthetic divisionFor an expression to be a factor of a polynomial, hence the remainder of the polynomial must be zero
Given the polynomial function x^3-2x^2-5x+6 divided by x + 2 according to the synthetic division
x^2 - 4x + 3
x + 2 | x^3-2x^2-5x+6
x^3 + 2x^2
------------------------------------
-4x^2 - 5x
-4x^2 - 8x
----------------------------------------
3x + 6
3x + 6
Since the remainder of the synthetic division is 0, hence x + 2 is a factor of the polynomial
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there's 240 candy bars 1/4 of candy bars are snickers 1/3 of the candy bars are twix 1/8 of the candy bars are hershey. how many candy bars are Mars? explain not with a lot of words but in numbers please.
Answers
Answer:
you have to add all the fractions of the candy1/4+1/3+1/8
=17/24
subtract from 1Step-by-step explanation:
1-17/24
=7/24
multiply with the total number of candy7/24×240
=70
Sam puts $500 in the bank and does not put any more money in or take any out for elght years. He receives 6% Interest
yearly, compounded semi-annually. How much money will be in the account after elght years?
O $793.44
$796.92
O $800.52
O $802.36
Answers
Answer:
$793.44
Step-by-step explanation:
If triangle QRS is dilated by a scale factor of 1/5 through the origin, which of the following points represents the coordinates of R’?
Answers
Answer:
-.4/.2
Step-by-step explanation:
take the points and divide them from 1/5
Dilation involves changing the size of triangle QRS
The coordinates of R' are (-2/5, 1/5)
The coordinate of R is given as:
\(\mathbf{R =(-2,1)}\)
The scale factor is given as 1/5; i.e.
\(\mathbf{k = \frac 15}\)
The dilation occurs through the origin,
So, the coordinates of R' is calculated using:
\(\mathbf{R' = k \times R}\)
This gives
\(\mathbf{R' = \frac 15 \times (-2,1)}\)
Multiply
\(\mathbf{R' = (-\frac 25,\frac 15)}\)
Hence, the coordinates of R' are (-2/5, 1/5)
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Sometimes Zach's teammates arrive late to basketball practice. The following dot plot shows how many times each teammate was late to practice last week.
Find the mean number of times Zach's teammates were late to practice last week.
BRAINLIEST!!!!!
Answers
Answer:
3.5.
Step-by-step explanation:
Zach's teammates were late to practice last week. According to Khan Academy the answer is 3.5.
The mean number of times Zach's teammates were late to practice last week is 3.5
What is dot plot?A dot plot visually groups the number of data points in a data set based on the value of each point.
Given that, the following dot plot shows how many times each teammate was late to practice last week.
We need to find mean,
Total elements = 2 + 4 + 4 + 4 = 14
Data points = 4
Mean = 14 / 4 = 3.5
Hence, the mean number of times Zach's teammates were late to practice last week is 3.5
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A ladder is leaning against a building, forming a 70° angle with the ground: The base of the ladder is 8.2 ft from the base of the
building.
What is the length of the ladder?
Round your answer to the nearest tenth of a foot.
22.5 ft
24.0 ft
28.0 ft
28.7 ft
Answers
The length of the ladder that is leaning on the building would be = 24ft. That is option B.
How to determine the length of the ladder?To determine the length of the ladder, the sine rule needs to be obeyed. That is
= a/sinA = b/sinB
Where;
a = 8.2 ft
A = 180-( 70+90
= 180- 160
= 20°
b = X
B = 90°
That is;
8.2/sin20° = b/sin90°
Make b the subject of formula;
b = 8.2×1/0.342020
= 23.9
= 24 ft
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